The number of pentacube triples, or sets of three different pentacubes, is C(29, 3), or 3654. Most of these triples can be joined to form symmetric 15-cubes. These 39 cannot: AFI, AIS/AIS′, AIT, ASX/AS′X, ATX, FGI/FG′I, GIK/G′IK, GKX/G′KX, HIX/H′IX, HIZ/H′IZ, IEV/IE′V, IJW/IJ′W, IJX/IJ′X, IKS/IKS′, IKU, IQT, IQZ, IRX/IR′X, JXZ/J′XZ, KLX, QTX, QXZ, RTX/R′TX, RXZ/R′XZ.
At the other extreme, the triple BLN can form 27086 different symmetric polycubes!
Just 23 triples have unique solutions. These triples are shown below. To see a solution, click on the triple. Cross-sections are shown from top to bottom.
|BIK||BIS (BIS′)||FGX (FG′X)||FGZ (FG′Z)|
|HIT (H′IT)||HXZ (H′XZ)||IJS (IJ′S′)||IJZ (IJ′Z)|
|IRT (IR′T)||IRZ (IR′Z)||KUW||QUX|
I am indebted to Gál Péter for his investigation of symmetric pentacube pairs.